nag_real_gen_lin_solve (f04bac)

+− Contents

1 Purpose

nag_real_gen_lin_solve (f04bac) computes the solution to a real system of linear equations AX=B, where A is an n by n matrix and X and B are n by r matrices. An estimate of the condition number of A and an error bound for the computed solution are also returned.

3 Description

The LU decomposition with partial pivoting and row interchanges is used to factor A as A=PLU, where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations AX=B.

5 Arguments

1:
order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:
order=Nag_RowMajor or Nag_ColMajor.

2:
n – IntegerInput

On entry: the number of linear equations n, i.e., the order of the matrix A.

Constraint:
n≥0.

3:
nrhs – IntegerInput

On entry:
the number of right-hand sides r, i.e., the number of columns of the matrix B.

On exit: if fail.code=NE_NOERROR, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row ipiv[i-1]. ipiv[i-1]=i indicates a row interchange was not required.

On exit: if fail.code=NE_NOERROR or NE_RCOND, the n by r solution matrix X.

8:
pdb – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if order=Nag_ColMajor,
pdb≥max1,n;

if order=Nag_RowMajor, pdb≥max1,nrhs.

9:
rcond – double *Output

On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A, computed as rcond=1/A1A-11.

10:
errbnd – double *Output

On exit: if fail.code=NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution x^, such that x^-x1/x1≤errbnd, where x^ is a column of the computed solution returned in the array b and x is the corresponding column of the exact solution X. If rcond is less than machine precision, then errbnd is returned as unity.

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_RCOND

A solution has been computed, but rcond is less than machine precision
so that the matrix A is numerically singular.

NE_SINGULAR

Diagonal element value of the upper triangular factor is zero.
The factorization has been completed, but the solution could not
be computed.

7 Accuracy

The computed solution for a single right-hand side, x^, satisfies an equation of the form

A+Ex^=b,

where

E1=OεA1

and ε is the machine precision. An approximate error bound for the computed solution is given by

x^-x1x1≤κAE1A1,

where κA=A-11A1, the condition number of A with respect to the solution of the linear equations. nag_real_gen_lin_solve (f04bac) uses the approximation E1=εA1 to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

8 Further Comments

The total number of floating point operations required to solve the equations AX=B is proportional to 23n3+n2r. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.